
Introductory concepts 11
x(t)=
Z
∞
−∞
X( f )e
−i2π ft
df (1.26)
X( f )=
Z
∞
−∞
x(t)e
i2π ft
dt (1.27)
1.4.3 Finite discrete signal
In practice we deal with discrete signals of finite duration. The Fourier transform
that operates on this kind of signal is called discrete Fourier transform (DFT) and the
algorithms that implement it are FFT (fast Fourier transform).
The DFT formula can be derived from (1.23). The signal to be transformed is
N samples long x = {x[0],...,x[n],...x[N −1]} and the samples are taken every T
s
seconds. It is assumed that the finite signal x is just one period of the infinite periodic
sequence with period T = N ·T
s
. The process of sampling can ...